Two spherical bodies of mass M and 5 M and radii R and 2 R are released in free space with initial separation between their centres equal to 12 R. If they attract each other due to gravitational force only, then the distance covered by the smaller body before collision is :
7.5 R
Let m1=M and m2=5M
Let centre of mass C at a distance x1 from m1 and x2 from m2.
m1x1=m2x2Mx1=5Mx2∴x1=5x2 and x1+x2=12R∴5x2+x2=12R∴6x2=12Rx2=2R∴x1=10R
Since the masses are moving under mutual attraction the position of centre of mass remains constant.
When the masses are in contact, let x′1 and x′2 be the distance of their centres from the centre of mass.
∴m2x′1=m2x′2∴Mx′1=5Mx′2∴x′1=5x′2Also x′1+x′2=3R5x′2+x′2=3R6x′2=3R∴x′2=0.5R and x′1=2.5R
Hence the distance travelled by the smaller mass is
x1 –x1’ = 10R – 2.5R = 7.5R